Rational Curves on Fano Varieties

Fano 品种的有理曲线

• 批准号: 2101935
• 负责人: Roya Beheshti Zavareh
• 金额: $ 20.39万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101935&HistoricalAwards=false
• 关键词: Rational; Curves; Fano; Varieties

Algebraic varieties are common zeros of collections of polynomial equations. Rational curves are the simplest algebraic varieties, and an important approach to study the geometry of algebraic varieties is to study parameter spaces of rational curves contained in them. These parameter spaces are themselves varieties with rich geometry, and their study has broad applications in higher dimensional algebraic geometry, enumerative geometry, arithmetic geometry, and questions inspired by mathematical physics. In this project, several open questions on various aspects of the geometry of spaces of rational curves on algebraic varieties are investigated. The project provides training opportunities for graduate students. The focus of the first part of the project is the study of spaces of rational curves (as well as rational surfaces and linear subvarieties) contained in hypersurfaces. Hypersurfaces of low degree in projective space form an important testing ground for the study of rationally connected and Fano varieties as well as several other questions in birational geometry. Despite some progress over the past few years, some of the basic properties of these spaces are still unknown. A major guiding question for the study of rational curves on hypersurfaces is which Fano hypersurfaces are rational or unirational. The second part of the project is on the study of rational curves on varieties from the perspective of Geometric Maninís conjecture which predicts the growth rate of a counting function associated to the irreducible components of moduli spaces of rational curves on a variety. In this part, several questions on the geometry of spaces of rational curves on Fano threefolds in characteristic zero and on del Pezzo surfaces over fields of finite characteristic are investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数簇是多项式方程集合的公共零点。有理曲线是最简单的代数簇，研究代数簇几何的一个重要途径是研究代数簇中有理曲线的参数空间。这些参数空间本身就是丰富的几何变体，它们的研究在高维代数几何、枚举几何、算术几何以及数学物理问题中有着广泛的应用。在这个项目中，几个开放的问题上的各个方面的几何空间的有理曲线的代数簇进行了调查。该项目为研究生提供培训机会。该项目第一部分的重点是研究超曲面中包含的有理曲线（以及有理曲面和线性子簇）的空间。射影空间中的低次超曲面是研究有理连通和Fano簇以及双有理几何中其他几个问题的重要实验场。尽管在过去几年中取得了一些进展，但这些空间的一些基本性质仍然是未知的。超曲面上有理曲线研究的一个主要指导问题是Fano超曲面是有理的还是单有理的。该项目的第二部分是从几何Maninís猜想的角度研究品种上的有理曲线，该猜想预测了与品种上有理曲线的模空间的不可约分量相关的计数函数的增长率。在这一部分中，几个问题的空间上的合理曲线的Fano三倍的特征零和del Pezzo曲面在外地的有限特征的调查。这一奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。